{"paper":{"title":"Homotopically Invisible Singular Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.DG","authors_text":"Andrei A. Agrachev, Antonio Lerario, Francesco Boarotto","submitted_at":"2016-03-29T20:03:44Z","abstract_excerpt":"Given a smooth manifold $M$ and a totally nonholonomic distribution $\\Delta\\subset TM$ of rank $d$, we study the effect of singular curves on the topology of the space of horizontal paths joining two points on $M$. Singular curves are critical points of the endpoint map $F:\\gamma\\mapsto\\gamma(1)$ defined on the space $\\Omega$ of horizontal paths starting at a fixed point $x$. We consider a subriemannian energy $J:\\Omega(y)\\to\\mathbb R$, where $\\Omega(y)=F^{-1}(y)$ is the space of horizontal paths connecting $x$ with $y$, and study those singular paths that do not influence the homotopy type of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08937","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}